[1]} suggest answers to these two questions by drawing a pre-proposal value \(y^{\prime }\)  from a symmetric proposal distribution on the tangent plane at \(x\) , denoted \(\mathcal {T}_x\) , and projecting this value down to the manifold to the proposal value \(y\) . The projection that [1]} define is not the typical \(l_2\)  projection, which would require a second-derivative of the constraint function to satisfy detailed balance, but rather a projection along the normal space of the manifold at \(x\) . This projection can be calculated using any non-linear solver that does not require a second derivative on \(g\) . With this scheme, the probability of taking a step on the manifold is calculated from the probability mass of the proposal distribution centered at the point \(x\) , defined on \(\mathcal {T}_x\) .
